Journal of Applied Mathematics

Volume 2013 (2013), Article ID 260856, 5 pages

http://dx.doi.org/10.1155/2013/260856

Research Article

Li-Yorke Sensitivity of Set-Valued Discrete Systems

Heng Liu,1,2 Fengchun Lei,¹ and Lidong Wang²

¹Department of Mathematics, Dalian University of Technology, Dalian, Liaoning 116600, China

²Department of Mathematics, Dalian Nationalities University, Dalian, Liaoning 116600, China

Received 6 October 2013; Accepted 27 November 2013

Academic Editor: Hongya Gao

Copyright © 2013 Heng Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Consider the surjective, continuous map and the continuous map of induced by , where is a compact metric space and is the space of all nonempty compact subsets of endowed with the Hausdorff metric. In this paper, we give a short proof that if is Li-Yoke sensitive, then is Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity of does not imply Li-Yorke sensitivity of .

1. Introduction

Throughout this paper a dynamical system is a pair where is a compact metric space with metric and is a surjective, continuous map.

The idea of sensitivity from the work [1, 2] by Ruelle and Takens was applied to topological dynamics by Auslander and Yorke in [3] and popularized later by Devaney in [4]. A system is called -sensitive if there exists a positive such that any is a limit of points satisfying the condition for some positive integer . According to Li and Yorke (see [5]), a subset is a scrambled set (for ), if any different points and from are proximal and not asymptotic; that is,

Li-Yoke sensitivity is introduced by Akin and Kolyada in [6]. A system is Li-Yorke sensitive if there exists such that every is a limit of points such that the pair is proximal but for any , and the positive is said to be a Li-Yorke sensitive constant of the system. A pair is -Li-Yorke sensitive if the pair is proximal but whose orbits are frequently at least apart.

A dynamical system is called spatiotemporal chaotic (see [6] or [7]) if every point is a limit point for points which are proximal to but not asymptotic to it. That is, for any and any open subset with , there is such that and are proximal and not asymptotic. It is easy to see that Li-Yorke sensitivity implies spatiotemporal chaos and sensitivity.

Román-Flores [8] and Fedeli [9] studied the interplay of chaos for discrete dynamical systems (individual chaos) with the corresponding set-valued versions (collective chaos). Recall that the map induced by on is defined by , . Then the pair is a dynamical system with the space endowed with the Hausdorff distance:

and . And various concepts of chaos in set-valued discrete systems have been researched recently (see [10–16]).

In this paper, we discuss the relationship between Li-Yorke sensitivity of and Li-Yorke sensitivity of . It will be shown that if is Li-Yoke sensitive, then is Li-Yorke sensitive. Furthermore, we give an example showing that Li-Yorke sensitivity of does not imply Li-Yorke sensitivity of . This paper discusses the further work of [17]. And by suing the obtained results, we give positive answers to Sharma and Nagar’s question in [18].

2. The Denjoy Homeomorphism and an Interval Map

Let be a compact metric space. For any nonempty subsets of and any , write , , and , where . When is a singleton, we write (resp., ) for (resp., ). For any nonempty subset of and any , write .

Write , where are nonempty subsets in . A subset is syndetic (or relative dense) if there is such that for every . A point is almost periodic if for any , is syndetic. A subset is thick if it contains arbitrarily long runs of positive integers. A dynamical system is transitive if for each pair of nonempty open subsets of , is nonempty. A point is transitive if the orbit is dense in . A system is minimal if any is transitive. We say is mixing if for each pair of nonempty open subsets , is cofinite, and is weakly mixing if is transitive. The set is said to be the -limit set of .

Lemma 1. If the system is minimal, then for any and any open subset , is syndetic. For some , if is an invariant closed set with , then for any , is thick.

Proof. For any and any open subset there are and such that . It is well known that if the system is minimal, then every is almost periodic. So is syndetic. Then is syndetic.

Since and is uniformly continuous, then for any and any , there is such that

So for some with , .

We will use as a model for the circle . The metric is defined by . Rigid rotation by the real number is then given by

Corresponding to the irrational , the Denjoy homeomorphism is an orientation preserving homeomorphism of the circle characterized by the following properties: the rotation number of is ; there is a Cantor set on which acts minimally; and if and are any two components of , then for some integer (see [19]). There is a Cantor function that semiconjugates with being a monotone surjection that collapses the components of (and so maps onto ) with .

Lemma 2. Let be the minimal subsystem of a Denjoy homeomorphism, and is a connected component of with . Then is -sensitive. Furthermore, for any and any , there is such that is syndetic.

Proof. For any and any , there is such that . Let be the arc in whose endpoints are and and whose length is . Then there exist and such that . Let be one of the connected components of with diam and . By Lemma 1, is syndetic. For any , . So .

Lemma 3. Let be the minimal subsystem of a Denjoy homeomorphism , and is a connected component of with . Then for any , there is such that for any with , .

Proof. Let be an arrangement of the connected components of with , , and . For any , has two elements at most. So for any , there is such that for any with , . For and , let be an arc, and . For the irrational , there exists such that and . So for any , there is such that . So . Let . Then .

Lemma 4 (see [17]). is not sensitive ( is a stable point).

Lemma 5 (see [6]). If a nontrivial system is weakly mixing then it is Li-Yorke sensitive.

Lemma 6. Let be the tent map which is . Then is Li-Yorke sensitive.

Proof. It is well known that the tent map is mixing [12]. Apply Lemma 5.

Example 7. is given by and which are the tent maps; is a constant mapping, and are linear where and is a transitive point of (see Figure 1).

Figure 1

Lemma 8. There is a positive , for any and any , there exists with such that the pair is -Li-Yorke sensitive.

Proof. By Lemma 6, is Li-Yorke sensitive. Let be a Li-Yorke sensitive constant of . Then for any , the lemma holds. For any and any , there exist an open interval with and such that . It is easy to see that is connected open neighborhood of . Because is Li-Yorke sensitive, there is such that is -Li-Yorke sensitive.

3. Li-Yorke Sensitivity

Lemma 9 (see [12]). Let be a system. Then the following statements are equivalent:

(i)is weakly mixing;(ii) is weakly mixing;(iii) is transitive.

Theorem 10. If a nontrivial system is weakly mixing, then is Li-Yorke sensitive.

Proof. By Lemma 9, is weakly mixing. Apply Lemma 5.

Theorem 11. If is Li-Yorke sensitive, then is Li-Yorke sensitive.

Proof. Let be Li-Yorke sensitive. There exists , for any and any , and there is a contract subset with (so, for any , ) such that is an -Li-Yorke sensitive pair of . So there exists a point with such that is an -Li-Yorke sensitive pair of .

4. A Counter Example

Example 12. Let be the minimal subsystem of a Denjoy homeomorphism, and , and let be the interval map given in Example 7. Let be a subset in polar coordinate system with metric defined by

And let the map be defined by . It is easy to see that is a dynamical system.

Proposition 13. is Li-Yorke sensitive.

Proof. For any , either or .

If , then there exists such that is a transitive point of and so . Since 2/9 is a fixed point of , by Lemma 1, is thick. By Lemma 2, for any , there exists such that is syndetic, so there is ; that is, . On the other hand, there is a sequence such that . So . So , is a -Li-Yorke sensitive pair.

If , by Lemma 8, there is a positive , for any , and there exists a point with such that is -Li-Yorke sensitive. It is not difficult to verify that is an -Li-Yorke sensitive.

To sum up, is a Li-Yorke sensitive constant of .

Proposition 14. is not sensitive.

Proof. Write . By Lemma 4, is a stable point of . So for any , there exists such that for every with and all , .

We will prove that is a stable point of . Let be the natural map defined by and be the natural map defined by . By the continuities of , , there exists such that for any with , and . Then for any ,

Proposition 15. is not spatiotemporal chaotic.

Proof. For any point , . By Lemma 3, there is such that for any with , . By the continuities of , there exists such that for any with , and . Let with . Then,

From Propositions 13 and 14 or from Propositions 13 and 15, we obtain the following at once.

Theorem 16. There is a dynamical system such that is Li-Yorke sensitive, but the set-valued discrete system induced by is not sensitive.

5. Li-Yorke Sensitivity of Interval Maps

Lemma 17 (see [20]). Let be a transitive interval map. Then one of the following conditions holds:

(i) is mixing;(ii)there is such that if and , in addition, is the unique fixed point of , and both and are mixing.

Theorem 18. Let be a transitive interval map. Then is Li-Yorke sensitive.

Proof. By Lemma 17, either is mixing or there is the unique fixed point such that and are mixing.

If is mixing, then is weakly mixing. Apply Lemma 5.

If there is the unique fixed point such that and are mixing, by Lemma 5, then and are Li-Yorke sensitive. It is easy to see that is Li-Yorke sensitive.

Example 19. is given by and which are the tent maps; is linear. It is easy to see that is Li-Yorke sensitive but is not transitive. So the converse version of Theorem 18 does not hold.

Example 20. is given by which is the tent map and which is linear. It is not difficult to get that is sensitive but is not Li-Yorke sensitive ( is a distal point).

The following example is an interval map which is spatiotemporal chaotic but is not Li-Yorke sensitive.

Example 21. is given by , and which are the tent maps, and , are linear, where , , (see Figure 2).

For any and any , there is such that . Since is mixing, there is such that is proximal but is not asymptotic. So is spatiotemporal chaotic.

On the other hand, for any , there exists such that for all . Since , for all , then any is not -unstable (i.e., there exists such that , for all ), so is not sensitive; especially, is not Li-Yorke sensitive.

Figure 2

Acknowledgments

This work is supported by NSFC 11001038 and the Fundamental Research Funds for the Central Universities DC120101112.

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